This paper explores the extension of dimension reduction (DR) techniques to the multi-dimension case by using the Einstein product. Our focus lies on graph-based methods, encompassing both linear and nonlinear approaches, within both supervised and unsupervised learning paradigms. Additionally, we investigate variants such as repulsion graphs and kernel methods for linear approaches. Furthermore, we present two generalizations for each method, based on single or multiple weights. We demonstrate the straightforward nature of these generalizations and provide theoretical insights. Numerical experiments are conducted, and results are compared with original methods, highlighting the efficiency of our proposed methods, particularly in handling high-dimensional data such as color images.

翻译：本文探索了利用爱因斯坦积将降维技术扩展到多维情形的方法。重点研究基于图的方法，涵盖线性和非线性方法，并涉及监督学习和无监督学习范式。此外，我们考察了线性方法的变体如斥力图和核方法。针对每种方法，我们提出基于单权重或多权重的两种推广形式，论证了这些推广的直观性并提供了理论见解。通过数值实验并与原始方法对比，结果表明所提出的方法在处理彩色图像等高维数据时具有显著优势。